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The Geometry of Kerr Black Holes (Dover Books on Physics), by Barrett O'Neill
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This unique monograph by a noted UCLA professor examines in detail the mathematics of Kerr black holes, which possess the properties of mass and angular momentum but carry no electrical charge. Suitable for advanced undergraduates and graduate students of mathematics, physics, and astronomy as well as professional physicists, the self-contained treatment constitutes an introduction to modern techniques in differential geometry.
The text begins with a substantial chapter offering background on the mathematics needed for the rest of the book. Subsequent chapters emphasize physical interpretations of geometric properties such as curvature, geodesics, isometries, totally geodesic submanifolds, and topological structure. Further investigations cover relativistic concepts such as causality, Petrov types, optical scalars, and the Goldberg-Sachs theorem. Four helpful appendixes supplement the text.
- Sales Rank: #205134 in Books
- Published on: 2014-03-19
- Released on: 2014-02-19
- Original language: English
- Number of items: 1
- Dimensions: 9.20" h x .70" w x 6.10" l, 1.20 pounds
- Binding: Paperback
- 400 pages
Most helpful customer reviews
40 of 41 people found the following review helpful.
An Invaluable Reference
By A Reader
In their 1279-page book "Gravitation," Misner, Thorne and Wheeler emphasize that the student of general relativity must master differential geometry on three different levels: (1) a pictorial level that reflects deep geometric intuition; (2) a conceptual level where equations may be expressed in coordinate-free or frame-independent notation; and (3) a computational level in local coordinates, which involves acquiring skill with the "debauch of indices" computations that are so characteristic of the subject, especially in the physics literature.
Barrett O'Neill is a highly accomplished differential geometer who worked in Riemannian geometry for some time before he began writing books on Lorentzian geometry and general relativity. All of his work is characterized by a mathematician's primary emphasis on the coordinate-free level (2), as mentioned in the preceding paragraph, before turning to local coordinate expressions. Mathematicians who are approaching general relativity as "outsiders," in particular, will find O'Neill's works extremely accessible---a welcome relief from the physics texts that are often written almost exclusively in index-based notation. O'Neill's book "Semi-Riemannian Geometry with Applications to Relativity" was written in 1983, and in my opinion it still remains the best introduction to Lorentzian geometry and general relativity for the well-prepared student who wants to see the mathematics "done right."
The book under review was first published in 1995, and it offered the first book-length treatment of Kerr spacetime written in a modern mathematical style, stressing both coordinate-free and coordinate-based computations. A casual comparison of O'Neill's book with Chandrasekhar's classic "The Mathematical Theory of Black Holes" (1983) will immediately reveal profound differences in the mathematical style of the two books. As Misner, Thorne and Wheeler said, the student who would master relativity theory must learn to read both styles of text with comfort.
The great virtue of O'Neill's books, however, is that they first provide profound conceptual insights into the more elusive concepts of general relativity through their elegant, coordinate-free expression. Once one has understood a particular concept, it is then relatively easy to explore its local coordinate expression; moving in the reverse direction can be quite difficult, however, especially for those who have not developed the intimate familiarity with complex index manipulations that comes from years of practice.
The fifth chapter of O'Neill's "The Geometry of Kerr Black Holes" contains an amazingly lucid discussion of the Weyl curvature tensor and its use in assigning Petrov Types to spacetimes. This chapter alone is worth the price of the entire book. Again, a comparison of this chapter with the related Chapter 4 in Stephani, Kramer, et. al.'s well-known "Exact Solutions to Einstein's Field Equations, 2nd Ed.," will
reveal sharp contrasts. O'Neill begins with a discussion of how the Hodge star operator * provides a complex vector space structure on the second exterior product of each tangent space; the Weyl curvature tensor induces an operator on this vector space that commutes with * and hence may be viewed as a complex linear operator. The Petrov classification emerges through a consideration of the complex eigenvalues of this linear operator. One might struggle for some time to apprehend these fundamental mathematical facts from the coordinate-based approach, which begins with ponderous eigenvalue equations written out in index notation and little or no discussion of the basic linear-algebraic concepts that underlie those equations.
In summary, O'Neill's book is highly recommended to the mathematician who is interested in general relativity, and to the physicist who desires to see the mathematics of GR expressed in both coordinate-free and coordinate-based formulas. The book stands in good company with related works by authors such as Theodore Frankel, Norbert Straumann, Rainer Sachs, and Jerrold Marsden, all of whom have written wonderful books on mathematical physics that emphasize the modern approach to differential geometry. It is regrettable that at the time of this review, O'Neill's book appears to have gone out of print.
19 of 20 people found the following review helpful.
A great book. Unique.
By J. Bielawski
This is the only book-length treatment of the subject. Besides the introductory chapters, it covers three topics in detail: maximal extensions, geodesics, and Petrov types/optical scalars/Newman-Penrose formalism/Goldberg-Sachs theorem. As the previous reviewer mentioned, it's mathematically very solid and it doesn't gloss over "obvious" details. For example, the extension of the Boyer-Lindquist formulas to the rotation axis is done explicitly. It's not particularly difficult but it's not a complete triviality either, and GR textbooks usually don't even mention the problem. (In the Schwarzschild case this is a non-issue thanks to the spherical symmetry.)
The maximal extensions chapter is a model of exposition, done in a very no-nonsense way (without Penrose diagrams - surprise!). The derivation of the Kruskal-Szekeres-like coordinates there and their extension over the "missing" crossing spheres are beautifully done. The chapter also includes remarks on Kerr isometries and certain topological tricks one can play there.
The chapter on geodesics in Kerr spacetime is the heart of the book. It's very detailed and makes wise pedagogical choices: it starts with the four motion constants (including the Carter constant) right away. The author then observes that the form of the resulting first-order ODEs implies that certain algebraic expressions must be nonnegative. This constraint alone yields a stupendous amount of information.
For example, to study the behaviour of the geodesic r-coordinate one examines regions of positivity inside a 4-dimensional parameter space (e,Q,L,r) (escape energy, Carter's constant, angular momentum, radial coordinate). It may seem daunting at first and I suspect O'Neill spent a few sleepless nights trying to come up with an elegant systematic presentation. His idea (I think it's his? Not sure if this approach exists elsewhere in the literature?) is to examine one (L,r) graph per each (e,Q) pair. Turns out the former then fall into four general patterns the author calls "continents", "barrier", "bay", and "lake". Examining those gives _tons_ of information about r-coordinates of geodesics.
I found a few small problems with the book, none of them fatal. In an ideal world they ought to be fixed but any reasonably careful reader can spot and repair them quickly:
1. There are many typos, esp. in Chapter 4 (Kerr geodesics). Nothing big, mostly wrong signs, upside down fractions, labels in figures, etc. Surprisingly, very few of those are in the complicated formulas for the metric of the maximally extended spacetime. [Edited later: these small typos are not terribly important except one: the curvature 2-forms for the metric. These formulas are likely to be referred to much more than all the others so I thought I'd point out that the second term of Omega^0_3 on page 98 should have a PLUS sign in front, not minus. (Yes, I got bitten by it!) So it's PLUS epsilon J omega^1 /\ omega^2.]
2. There is a strange small mistake regarding the spherical theta coordinate. The author says (p. 184) "theta is globally defined on any Kerr spacetime and, though not smooth at the poles, has well-defined directional derivatives v[theta] there". This is not true! The coordinate theta in terms of (x,y) is equal to arcsin(sqrt(x^2+y^2)) so it doesn't have partial derivatives at (0,0). Fortunately, in that context only the derivative of theta _squared_ is used, and that function does extend to the poles. Almost everywhere else when the theta coordinate is used, it actually refers to the theta coordinate _of a geodesic_, and this function is presumed continuously extended beyond the usual range [0, pi] as needed (so this theta function _is_ differentiable at the poles). The above distinction must be understood before reading the book.
There is a similar incorrect remark on p. 44 where the author says sin(theta) is smooth on the 2D sphere S^2. It isn't, but its square is, and again the sin-square is what's used in the book. The cosine is used without the square but fortunately it _is_ smooth at the poles.
3. It looks like a bit of material is missing in the proof of geodesic completeness. Perhaps a victim of a last minute cut and paste? Proposition 4.3.9 doesn't seem to follow from what went on before, which was: under certain technical restrictions a geodesic can be extended past certain points. It's not clear how to remove those technical restrictions (stated in Lemma 4.3.5). I think I know how to fix this (looks like the key is that simple zeros of r are isolated) but a few extra lines of author's explanation would go a long way.
[Edited later: the "missing" bit would be just a sentence or two directing the reader to Proposition 4.3.3(2) which is a formula for the increase in the geodesic parameter in terms of abs. values of certain integrals of R and Theta. Since the geodesic r coordinate must bounce back and forth between a pair of (simple) zeros of R, and R is fixed, the increase of the parameter must be uniformly bounded from below for each bounce. In other words, the parameter must go to infinity as the geodesic keeps getting extended.]
4. In the proof of Proposition 4.8.4, Case 3 is incomplete. The problem is the fourth zero of R(r) is guaranteed to be greater than 2M only for L large enough. This can be seen clearly on Fig. 4.9: inside the region N for L low enough there is an "overhang" and we cannot just move to the right while staying inside N. Again, this is easy to fix, one just needs to move the point (r_i, L_i) up while remaining inside N in order to make L large enough so the above condition (r>2M) can be applied. One way to do it is to move the point (r_i, L_i) vertically (either up or down) until it meets the midline L_v(r). Then move along the midline upward until L is large enough.
None of this is important, it's a GREAT book. It's very good of Dover to bring it back. Super highly recommended!
1 of 1 people found the following review helpful.
Good intermediate book!
By Amazon Customer
I am familiar with basic General Relativity and tensors. This book picks up where I left off; covering in sufficient detail the various coordinate (and non coordinate) maps/covers of rotating black holes. It was written in the 90's so it is probably a little dated; but the level is perfect for me. I am going to try to implement the formula in Sage's manifold module.
The first chapter introduces the needed mathematical/differential geometry structures. You need to have some background though.
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